Method and system for determining multi-scale spatial variation of soil structure

Abstract

The invention discloses a determination method and system for multi-scale spatial variation of a soil structure. The determination method comprises the following steps: collecting a soil sample; calculating a soil volume fractal dimension based on the soil sample; calculating a global Moran index and a local Moran index based on the soil sample and the soil volume fractal dimension, and obtaining spatial autocorrelation; checking whether the spatial correlation accords with normal distribution or not by using a single sample K-S, and obtaining a checking result; calculating a variation function according to the test result, and obtaining an optimal fitting model according to a calculation result; and based on the optimal fitting model, performing interpolation by using Kriging interpolation to obtain the spatial distribution condition of the fractal dimensions of the soil. According to the invention, the requirements of continuous monitoring and multi-scale comprehensive analysis on the dynamic change of the soil structure can be met.

Description

Technical Field

[0001] The present invention belongs to the technical field of quantifying the spatial variability of basic soil properties, and particularly relates to a method and system for determining the multi-scale spatial variability of soil structure. Background Art

[0002] Quantifying soil structure is of great significance for understanding the dynamic mechanisms of the surface Earth system and ecological environment processes, because soil structure fundamentally determines the key parameters of the surface Earth system, affects other soil properties, including the hydrodynamic characteristics of the vadose zone, solute transport characteristics, and thermodynamic properties, and controls the material and energy migration and transformation processes in the saturated-unsaturated zone.

[0003] Traditional soil structure research mainly focuses on a single scale or local range, and it is difficult to comprehensively reflect the multi-scale spatial heterogeneity of soil structure. Although modern multi-scale research methods, such as remote sensing monitoring and fractal dimension analysis, etc., provide possibilities for soil structure research at large scales or micro scales, there are still many challenges in practical applications, such as: difficult data acquisition, high computational complexity, insufficient model universality, etc.

[0004] At present, the measurement and analysis of soil multi-scale spatial structure mainly rely on theories and methods such as geostatistics, remote sensing image analysis, and fractal geometry. However, existing technologies often have difficulty in accurately characterizing the spatial variability characteristics of soil structure at multiple scales, and there are also limitations in terms of computational accuracy, operability, and adaptability. Therefore, there is an urgent need for a method and system for determining the multi-scale spatial variability of soil structure to meet the needs of continuous monitoring and multi-scale comprehensive analysis of the dynamic changes of soil structure. Summary of the Invention

[0005] To solve the above technical problems, the present invention proposes a method and system for determining the multi-scale spatial variability of soil structure, which can meet the needs of continuous monitoring and multi-scale comprehensive analysis of the dynamic changes of soil structure.

[0006] The present invention provides a method for determining the multi-scale spatial variability of soil structure, including:

[0007] Collecting soil samples;

[0008] Calculating the soil volume fractal dimension based on the soil samples;

[0009] Calculating the global Moran's I and local Moran's I based on the soil samples and the soil volume fractal dimension to obtain spatial autocorrelation;

[0010] Using a one-sample K-S test to check whether the spatial correlation conforms to a normal distribution to obtain a test result;

[0011] Calculate the variogram based on the inspection results, and obtain the optimal fitting model according to the calculation results;

[0012] Interpolate using Kriging interpolation based on the optimal fitting model.

[0013] Optionally, the soil samples collected include:

[0014] Close-range sampling of small-scale profiles and long-range sampling of large-scale regions.

[0015] Optionally, calculating the soil volume fractal dimension based on the soil samples includes:

[0016] Calculate the particle size and corresponding volume data of the soil samples based on the soil samples;

[0017] Based on the particle size and corresponding volume data of the soil samples, perform linear fitting of its slope using the least squares method and calculate the soil volume fractal dimension.

[0018] Optionally, calculating the global Moran's I and local Moran's I based on the soil samples and soil volume fractal dimension to obtain spatial autocorrelation includes:

[0019] For small-scale profiles, calculate the spatial autocorrelation of the entire profile using the global Moran's I and the spatial autocorrelation of each sampling point using the local Moran's I;

[0020] For large-scale regions, calculate the spatial autocorrelation of the entire region using the global Moran's I and calculate the spatial autocorrelation of each profile using the local Moran's I.

[0021] Optionally, calculating the overall spatial correlation of the large-scale region or small-scale profile using the global Moran's I includes:

[0022]

[0023] where n is the number of sample points to be estimated, x i and x j are the observed values of the variable x at sample points i and j respectively, is the average value of x, W ij is the adjacent weight;

[0024] When I = 0, it indicates that there is no correlation in the spatial distribution of the variable and it is randomly distributed;

[0025] When I approaches 1, the spatial distribution relationship between variables shows complete positive correlation and aggregation;

[0026] When I approaches -1, the variables show complete negative correlation and dispersed distribution.

[0027] Optionally, calculating the spatial correlation for each profile at a large scale or each sampling point at a small scale using the local Moran's index includes:

[0028]

[0029] where: I i is the local Moran's index of the i-th observation point, x i and x j are the variable values of observation points i and j, respectively, is the average of the variable values of all observation points, w ij is an element in the spatial weight matrix, representing the spatial relationship between observation points i and j, usually a binary value, S 0 is the sum of all non-diagonal elements of the spatial weight matrix, that is Here is because each pair is calculated twice. When I i is significantly positive, it indicates the existence of clusters of similar high or low values; when I i is significantly negative, it may mean there is heterogeneity between this point and the surrounding points.

[0030] Optionally, using the one-sample K-S test to check whether the spatial correlation conforms to a normal distribution, and obtaining the test results includes:

[0031] Using the one-sample K-S test to perform a normal distribution test on the soil fractal dimension value at a 5% confidence level to obtain the test results.

[0032] Optionally, according to the test results, calculating the variogram includes:

[0033] If it conforms to a normal distribution, calculate the variogram;

[0034] If it does not satisfy the normal distribution, perform one to three transformations on the soil samples. If the soil samples still do not satisfy the normal distribution, increase the collection of soil samples until the test results conform to the normal distribution.

[0035] Optionally, the method for calculating the variogram is:

[0036]

[0037] where γ(h) is the sample semivariogram value, Z(x i ) is the sample data series, N h is the number of sample pairs of the sample semivariogram value.

[0038] Optionally, the method for interpolation using Kriging interpolation is:

[0039]

[0040] Among them, z(x 0 ) is the estimated value of point x 0 , z(x i ) is the measured value of the known point x i , n is the number of sample points within the estimated search radius, and λ i is the weight coefficient.

[0041] The present invention also provides a determination system for multi-scale spatial variability of soil structure, which is characterized by including: a data acquisition module, a data processing module, a data verification module, and a data interpolation module;

[0042] The data acquisition module is used to collect soil samples;

[0043] The data processing module is used to calculate the soil volume fractal dimension based on the soil samples, calculate the global Moran's I and local Moran's I based on the soil samples and the soil volume fractal dimension, and obtain the spatial autocorrelation;

[0044] The data verification module is used to use the one-sample K-S test to check whether the spatial correlation conforms to the normal distribution and obtain the test result;

[0045] The data interpolation module is used to calculate the variogram according to the test result, obtain the optimal fitting model according to the calculation result, and perform interpolation using Kriging interpolation based on the optimal fitting model.

[0046] Compared with the prior art, the present invention has the following advantages and technical effects:

[0047] The present invention uses the soil volume fractal dimension D combined with the K-S test, the global Moran's I and local Moran's I, the variogram, the fitting model, and Kriging interpolation to evaluate the multi-scale spatial variability of soil structure. It can not only quantitatively analyze the structural characteristics of small-scale and large-scale soils, but also reveal the multi-scale spatial variability degree of soil structure; it solves the limitations existing in the calculation accuracy, operability, and adaptability in the prior art. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] The drawings constituting a part of this application are used to provide a further understanding of this application. The schematic embodiments of this application and their descriptions are used to explain this application and do not constitute an improper limitation to this application. In the drawings:

[0049] Figure 1 is a flowchart of a method for determining multi-scale spatial variability of soil structure according to an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0050] It should be noted that, without conflict, the embodiments in the present application and the features in the embodiments may be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0051] It should be noted that the steps shown in the flowchart of the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although the logical order is shown in the flowchart, in some cases, the steps shown or described can be executed in a different order than here.

[0052] This embodiment proposes a method for determining the multi-scale spatial variability of soil structure, as Figure 1 shown, which specifically includes the following steps:

[0053] Collect soil samples;

[0054] Based on the soil samples, calculate the soil volume fractal dimension;

[0055] Based on the soil samples and the soil volume fractal dimension, calculate the global Moran's I and local Moran's I to obtain the spatial autocorrelation;

[0056] Use the one-sample Kolmogorov-Smirnov (K-S) test to check whether the spatial correlation conforms to the normal distribution and obtain the test result;

[0057] According to the test result, calculate the variogram, and according to the calculation result, obtain the optimal fitting model;

[0058] Based on the optimal fitting model, perform interpolation using Kriging interpolation.

[0059] 1. Specifically, collect soil samples from the study area and collect soil samples from different profiles;

[0060] 2. Obtain the particle size distribution of the soil samples and calculate the soil volume fractal dimension D based on the soil particle volume fractal formula;

[0061] 3. Calculate the global Moran's I and local Moran's I to evaluate the spatial autocorrelation of the soil profile;

[0062] 4. Use the one-sample Kolmogorov-Simirnov (K-S) test to perform a normal distribution test on the soil fractal dimension data;

[0063] 5. Apply the geostatistical semi-variogram function to analyze the spatial variability of the soil fractal dimension;

[0064] 6. Interpolate the multi-scale variability of the soil structure by the Kriging interpolation method.

[0065] Furthermore, collecting soil samples includes:

[0066] Close-range sampling of small-scale profiles and long-range sampling of large-scale regions.

[0067] Specifically, soil samples from different profiles are collected. For small-scale profiles, only single or adjacent soil profiles can be collected, and for large-scale profiles, soil profiles at a greater distance are collected; multiple soil samples are taken in the depth direction for each profile.

[0068] Furthermore, based on the soil samples, calculating the soil volume fractal dimension includes:

[0069] Based on the soil samples, calculate the particle size and corresponding volume data of the soil samples;

[0070] Based on the particle size and corresponding volume data of the soil samples, use the least squares method to fit the slope of a straight line and calculate the soil volume fractal dimension.

[0071] Specifically, obtain the particle size distribution of the soil samples;

[0072] Analyze the measured soil particle volume distribution using the soil particle volume fractal formula to obtain the soil particle volume fractal dimension;

[0073] The calculation of soil particle size fractal is based on the fractal dimension model of different particle size volume distributions and can be expressed as:

[0074]

[0075] where d i (i = 1, 2, 3..., n) is the soil particle diameter, d max is the maximum soil particle diameter, D is the fractal dimension, δ is the particle size variable of the soil particles, V(δ < d i ) is the volume of soil particles with a particle size less than d i , and V0 is the volume of the soil sample.

[0076] When calculating the fractal dimension, first calculate and from the measured particle size and its volume data, and then use as the ordinate and as the abscissa, and use the least squares method to fit the slope of a straight line and calculate the fractal dimension D of the sample.

[0077] Furthermore, based on the soil samples and the soil volume fractal dimension, calculate the global Moran's I and local Moran's I to obtain the spatial autocorrelation, including:

[0078] For small-scale profiles, use the global Moran's I to calculate the spatial autocorrelation of the entire profile, and the local Moran's I to calculate the spatial autocorrelation of each sampling point;

[0079] For large-scale regions, the global Moran's I index is used to calculate the spatial autocorrelation of the entire region, and the local Moran's I index is used to calculate the spatial autocorrelation of each profile.

[0080] Specifically, the global Moran's I index is adopted to evaluate the spatial autocorrelation degree of all observations or the whole of a single soil profile in the entire study area, and the local Moran's I index is used to identify the spatial autocorrelation characteristics of specific soil sampling points and their surrounding areas.

[0081] The formula for the global Moran's I index is as follows:

[0082]

[0083] In the formula: where n is the number of sample points to be estimated, xi and xj are the observed values of variable x at sample points i and j respectively, is the average value of x, and W ij is the adjacent weight;

[0084] The value range is [-1, 1]. When I = 0, it indicates that there is no correlation in the spatial distribution of the variable and it shows a random distribution. When I approaches 1, the spatial distribution relationship between variables presents a completely positive correlation and an aggregated state. When I approaches -1, the variables show a completely negative correlation and a dispersed distribution.

[0085] The formula for the local Moran's I index is as follows:

[0086]

[0087] where: I i is the local Moran's I index of the ith observation point. x i and x j are the variable values of observation points i and j respectively. is the average of the variable values of all observation points. w ij is an element in the spatial weight matrix, indicating the spatial relationship (such as adjacent or not) between observation points i and j, usually a binary value (1 means adjacent, 0 means not adjacent) or a distance-based weight. S 0 is the sum of all non-diagonal elements of the spatial weight matrix, that is Here is because each pair is calculated twice. When I i is significantly positive, it indicates the existence of similar high-value or low-value aggregations; when I i is significantly negative, it may mean the existence of heterogeneity between this point and the surrounding points.

[0088] Furthermore, the one-sample K-S test is used to check whether the spatial correlation conforms to the normal distribution, and the obtained test results include:

[0089] The normal distribution test of the soil fractal dimension value is carried out using the one-sample Kolmogorov-Smirnov (K-S) test at a 5% confidence level to obtain the test results.

[0090] Furthermore, according to the test results, calculating the variogram includes:

[0091] If it conforms to the normal distribution, calculate the variogram;

[0092] If it does not satisfy the normal distribution, perform one to three transformations (such as logarithmic transformation or Box-Cox transformation, etc.) on the soil samples. If the soil samples still do not satisfy the normal distribution, increase the collection of soil samples until the test results conform to the normal distribution.

[0093] Specifically, the one-sample Kolmogorov-Smirnov (K-S) test is used to conduct the normal distribution test of the soil fractal dimension D value at a 5% confidence level, and conclusions are drawn based on the magnitude of the P value;

[0094] If the K-S test result conforms to the normal distribution, continue with the subsequent operations. If it does not satisfy the normal distribution, one or more logarithmic transformations or Box-Cox transformations, etc., can be performed on the data. If the data still does not satisfy the normal distribution, considering increasing the collection of soil samples;

[0095] The semivariogram in geostatistics is used to analyze and describe the spatial distribution, variation, and correlation characteristics of the soil fractal dimension;

[0096] The sample semivariogram γ(h) can well reflect the influence of random factors on the soil fractal dimension and the spatial correlation relationship. The sample semivariogram γ(h) can be expressed as:

[0097]

[0098] In the formula, γ(h) is the sample semivariogram value, Z(x i ) is the sample data series, N h is the number of sample pairs of the sample semivariogram value.

[0099] Use spherical models, exponential models, Gaussian models, linear models, etc. to fit the anisotropic semivariogram. Based on the principle that the model has a large coefficient of determination (R 2 ) and a small sum of squared residuals (RSS), range, and nugget value, select the optimal theoretical model and use it for Kriging interpolation;

[0100] Ordinary Kriging interpolation is performed based on the fitted model. Its expression is as follows:

[0101]

[0102] In the formula, z(x0 ) is the estimated value of x 0 , z(x i ) is the measured value of the known point x i , n is the number of sample points within the estimated search radius, and λ i is the weight coefficient;

[0103] The present invention also provides a determination system for multi-scale spatial variability of soil structure, which is characterized by including: a data acquisition module, a data processing module, a data verification module, and a data interpolation module;

[0104] The data acquisition module is used to collect soil samples;

[0105] The data processing module is used to calculate the soil volume fractal dimension based on the soil samples, calculate the global Moran's I and local Moran's I based on the soil samples and the soil volume fractal dimension, and obtain the spatial autocorrelation;

[0106] The data verification module is used to use the one-sample K-S test to check whether the spatial correlation conforms to the normal distribution and obtain the test result;

[0107] The data interpolation module is used to calculate the variogram according to the test result, obtain the optimal fitting model according to the calculation result, and perform interpolation using Kriging interpolation based on the optimal fitting model.

[0108] The following elaborates on this embodiment in conjunction with the accompanying drawings:

[0109] This embodiment provides a method for determining multi-scale spatial variability of soil structure, specifically as Figure 1 shown, including:

[0110] Step 1: Collection of soil samples in the study area. Collect multiple soil profiles, and sample at each depth, such as sampling every 10 cm or 20 cm, and soil samples of several meters can be taken to discuss the spatial variability of soil structure with depth.

[0111] Step 2: Obtain the particle size distribution of the soil samples, and calculate the soil volume fractal dimension D according to the soil particle volume fractal formula.

[0112] Step 3: For small-scale profiles, the global Moran's I is used to calculate the spatial autocorrelation of the entire profile, and the local Moran's I can calculate the spatial autocorrelation of each soil sampling point; for large-scale regions, calculations are performed by depth stratification. For example, the average values of the soil fractal dimension D values of 0-50 cm, 50-100 cm, and 100-150 cm are taken respectively, and the global Moran's I is used to calculate the spatial autocorrelation of 0-50 cm, 50-100 cm, and 100-150 cm in the entire study area; the local Moran's I calculates the spatial autocorrelation of individual profiles of 0-50 cm, 50-100 cm, and 100-150 cm.

[0113] Step 4: The K-S test is used to determine whether the soil fractal dimension data conforms to the normal distribution. For small scales, it is the data of individual profiles; for large scales, calculations are performed by depth stratification. For example, it is determined whether the mean data of all 0-50 cm, 50-100 cm, and 100-150 cm conform to the normal distribution respectively.

[0114] Step 5: If it meets the normal distribution, the subsequent steps are continued. If it does not meet the normal distribution, the data needs to be logarithmically transformed or transformed by Box-Cox etc. to make it meet the normal distribution. If it still does not meet the normal distribution after multiple transformations, denser sampling needs to be considered.

[0115] Step 6: The variogram of the entire profile is calculated for small scales, and the variogram of the sampling profiles in the entire study area is calculated by depth stratification for large scales. Exponential models, linear models, Gaussian models, etc. are used for fitting, and the model with the best fitting effect is selected according to the model evaluation index.

[0116] Step 7: The soil volume fractal dimension D value of the entire profile is interpolated by Kriging interpolation for small scales, and the soil volume fractal dimension D values of 0-50 cm, 50-100 cm, and 100-150 cm in the entire study area are interpolated by Kriging interpolation for large scales.

[0117] The following are specific embodiments of the present invention:

[0118] Taking the Daqing River and Ziya River as the study area, ten soil profiles are selected in the large-scale region. Soil is taken every 10 cm for each profile, and a total of 3 m of soil samples are taken. Each profile has a total of 30 soil samples.

[0119] Obtain the particle size distribution of all soil samples and calculate the soil volume fractal dimension D value.

[0120] In the depth direction, it is divided into three layers of 0-1 m, 1-2 m, and 2-3 m, and the D mean value of each profile is calculated by regarding 0-3 m as a whole.

[0121] The global Moran's index is used to calculate the spatial autocorrelation of the entire area of 0-1m, 1-2m, 2-3m, and 0-3m, and the local Moran's index is used to calculate the spatial autocorrelation of a single profile of 0-1m, 1-2m, 2-3m, and 0-3m.

[0122] The K-S test is used to determine whether all the mean data of 0-1m, 1-2m, 2-3m, and 0-3m satisfy the normal distribution.

[0123] The variograms of 0-1m, 1-2m, 2-3m, and 0-3m in the entire study area are calculated and fitted by different models in layers, and the optimal fitting model for each layer is selected according to the model evaluation index.

[0124] Kriging interpolation is used to perform spatial interpolation on the data of 0-1m, 1-2m, 2-3m, and 0-3m to obtain a continuous regional D-value distribution, which facilitates subsequent spatial variability analysis of soil structure.

[0125] The above is only a preferred specific implementation manner of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed in the present application should be covered by the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A method for determining multi-scale spatial variation of soil structure, characterized in that: include: Collect soil samples; Based on the soil sample, calculating the soil volume fractal dimension; Based on the soil sample and the soil volume fractal dimension, a global Moran's index and a local Moran's index are calculated to obtain spatial autocorrelation; Use single-sample KS to test whether the spatial correlation conforms to the normal distribution and obtain the test result; Calculate the variance function according to the test result, and obtain the best fitting model according to the calculation result; Based on the optimal fitting model, Kriging interpolation is used to obtain the spatial distribution of soil fractal dimension.

2. A method for determining multi-scale spatial variation of soil structure according to claim 1, characterized in that: Soil sampling includes: Close sampling of small-scale sections and long-distance sampling of large-scale areas.

3. A method for determining multi-scale spatial variation of soil structure according to claim 2, characterized in that: Based on the soil sample, calculating the soil volume fractal dimension includes: Based on the soil sample, calculating the particle size and corresponding volume data of the soil sample; Based on the particle size and corresponding volume data of the soil sample, the slope of the straight line is fitted using the least squares method, and the soil volume fractal dimension is calculated.

4. A method for determining multi-scale spatial variation of soil structure according to claim 2, characterized in that: Based on the soil sample and the soil volume fractal dimension, the global Moran index and the local Moran index are calculated to obtain the spatial autocorrelation including: For small-scale profiles, the global Moran's index is used to calculate the spatial autocorrelation of the entire profile, and the local Moran's index is used to calculate the spatial autocorrelation of each sampling point; For large-scale areas, the global Moran's index is used to calculate the spatial autocorrelation of the entire area, and the local Moran's index is used to calculate the spatial autocorrelation of each profile.

5. A method for determining multi-scale spatial variation of soil structure according to claim 4, characterized in that: Calculating the spatial correlation of the large-scale region or small-scale profile using the global Moran index includes: Where n is the number of sample points to be estimated, xi and xj are the observed values ​​of variable x at sample points i and j respectively. is the average value of x, W ij is the adjacent weight; When I = 0, it means that the variables have no correlation in this spatial distribution and are randomly distributed; When I tends to 1, the spatial distribution relationship between variables shows a completely positive correlation and aggregation state; When I tends to -1, the variables are completely negatively correlated and dispersed; The calculation of the spatial correlation of each large-scale profile or each small-scale sampling point using the local Moran index includes: Where: I i is the local Moran index of the ith observation point, x i and x j are the variable values ​​of observation points i and j respectively, is the average of all observed variable values, w ij is an element in the spatial weight matrix, representing the spatial relationship between observation points i and j, and S0 is the sum of all off-diagonal elements of the spatial weight matrix, that is, Here is because each pair is counted twice, when I i When it is significantly positive, it indicates that there is a similar high or low value clustering; when I i When it is significantly negative, it may mean that there is heterogeneity between the point and the surrounding points.

6. A method for determining multi-scale spatial variation of soil structure according to claim 2, characterized in that: The single-sample KS is used to test whether the spatial correlation conforms to the normal distribution. The test results include: The normal distribution test of the soil fractal dimension value is performed using a single sample KS test at a 5% confidence level to obtain the test result.

7. A method for determining multi-scale spatial variation of soil structure according to claim 2, characterized in that: According to the test results, calculating the variance function includes: If it conforms to the normal distribution, the variance function is calculated; If the normal distribution is not satisfied, the soil sample is transformed one to three times. If the soil sample still does not satisfy the normal distribution, the collection of soil samples is increased until the test results conform to the normal distribution.

8. A method for determining multi-scale spatial variation of soil structure according to claim 7, characterized in that: The method to calculate the variogram is: Among them, γ(h) is the sample semivariance function value, Z(x i ) is the sample data series, N h is the number of sample pairs of sample semivariance function values.

9. A method for determining multi-scale spatial variation of soil structure according to claim 1, characterized in that: The method of interpolation using Kriging interpolation is: Among them, z(x0) is the estimated value at x0, z(x i ) is a known point x i The measured value of n is the number of sample points within the estimated search radius, and λ i is the weight coefficient.

10. A system for determining multi-scale spatial variation of soil structure, characterized in that: include: Data acquisition module, data processing module, data verification module and data interpolation module; The data acquisition module is used to collect soil samples; The data processing module is used to calculate the soil volume fractal dimension based on the soil sample, calculate the global Moran index and the local Moran index based on the soil sample and the soil volume fractal dimension, and obtain the spatial autocorrelation; The data verification module is used to use a single sample KS to test whether the spatial correlation conforms to a normal distribution and obtain a test result; The data interpolation module is used to calculate the variance function according to the test result, obtain the best fitting model according to the calculation result, and perform interpolation based on the best fitting model using Kriging interpolation.

Images